Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 11F / iii

Table of contents
- Solution iii

Solution

Let

m_{j}

be the expected remaining time when the walk is at

j

. The first-step recurrence for gambler's ruin is

m_{j} = 1 + \frac{1}{2} m_{j - 1} + \frac{1}{2} m_{j + 1}, - a < j < b,

with

m_{- a} = m_{b} = 0

. The quadratic

m_{j} = (j + a) (b - j)

satisfies both the recurrence and the boundary conditions, and their finite linear system has a unique solution. Starting from zero therefore gives

E (T) = m_{0} = ab .

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